Geometric Sequence Calculator

About Geometric Sequence Calculator

Welcome to our Geometric Sequence Calculator, a powerful mathematical tool that calculates the nth term, sum of the first n terms, and infinite sum of any geometric sequence. Whether you are studying mathematics, preparing for exams, or solving real-world problems involving exponential growth or decay, this calculator provides accurate results with detailed step-by-step solutions and interactive visualizations.

What Is a Geometric Sequence?

A geometric sequence (also called a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). This multiplicative pattern distinguishes geometric sequences from arithmetic sequences, where terms differ by a constant addition.

For example, the sequence 3, 6, 12, 24, 48, ... is geometric because each term is twice the previous term (r = 2). The sequence 100, 50, 25, 12.5, ... is also geometric with r = 0.5, showing how terms can decrease.

Key Components of a Geometric Sequence

• First term (a₁): The starting value of the sequence
• Common ratio (r): The constant multiplier between consecutive terms
• nth term (aₙ): Any specific term at position n in the sequence
• Sum (Sₙ): The total of the first n terms

Geometric Sequence Formulas

The nth Term Formula

To find any term in a geometric sequence, use the formula:

Where a₁ is the first term, r is the common ratio, and n is the position of the term. The exponent is (n-1) because we multiply by r zero times to get the first term, once to get the second term, and so on.

Sum of First n Terms

The sum of the first n terms depends on whether the common ratio equals 1:

When r = 1, all terms are equal, so Sₙ = n × a₁.

Infinite Sum (Convergent Series)

When |r| < 1, the terms approach zero and the infinite sum converges:

If |r| ≥ 1, the series diverges and has no finite sum.

How to Use This Calculator

1. Enter the first term (a₁): Input the starting value of your geometric sequence. This can be positive, negative, or a decimal.
2. Enter the common ratio (r): Input the value by which each term is multiplied. The ratio can be positive, negative, or fractional.
3. Enter n: Specify which term position you want to find, and how many terms to sum.
4. Select precision: Choose the number of decimal places for your results (10-100).
5. Click Calculate: View the nth term, sum, sequence visualization, and step-by-step solution.

Understanding Sequence Behavior

Growth vs Decay

• Growth (r > 1): Terms increase without bound. Example: 2, 6, 18, 54, ... (r = 3)
• Decay (0 < r < 1): Terms decrease toward zero. Example: 100, 50, 25, ... (r = 0.5)
• Oscillating (-1 < r < 0): Terms alternate signs and decrease in magnitude. Example: 8, -4, 2, -1, ... (r = -0.5)
• Oscillating Growth (r < -1): Terms alternate signs and increase in magnitude. Example: 2, -6, 18, -54, ... (r = -3)
• Constant (r = 1): All terms equal the first term. Example: 5, 5, 5, 5, ...
• Alternating Constant (r = -1): Terms alternate between +a₁ and -a₁. Example: 7, -7, 7, -7, ...

Real-World Applications

Frequently Asked Questions

What is a geometric sequence?

A geometric sequence (or geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, 2, 6, 18, 54, ... is a geometric sequence with first term a₁=2 and common ratio r=3.

What is the formula for the nth term of a geometric sequence?

The nth term of a geometric sequence is given by the formula: aₙ = a₁ × r^(n-1), where a₁ is the first term, r is the common ratio, and n is the position of the term you want to find. For example, if a₁=3 and r=2, the 5th term is a₅ = 3 × 2^4 = 48.

How do you find the sum of a geometric sequence?

The sum of the first n terms of a geometric sequence is Sₙ = a₁(1-rⁿ)/(1-r) when r≠1, or Sₙ = n×a₁ when r=1. For an infinite geometric series where |r|<1, the sum converges to S∞ = a₁/(1-r).

When does a geometric series converge?

A geometric series converges (has a finite sum to infinity) when the absolute value of the common ratio is less than 1 (|r| < 1). This means terms get progressively smaller and approach zero. If |r| ≥ 1, the series diverges and has no finite sum.

What is the difference between geometric and arithmetic sequences?

In an arithmetic sequence, each term differs from the previous by a constant amount (common difference). In a geometric sequence, each term is a constant multiple (common ratio) of the previous term. Arithmetic: 2, 5, 8, 11 (add 3). Geometric: 2, 6, 18, 54 (multiply by 3).

Additional Resources

• Geometric Sequences - Mathematics LibreTexts
• Geometric Progression - Wikipedia
• Geometric Series - Wikipedia

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